A train is traveling at a
velocity near the speed of light. A man raiding in the middle of the car (Observer
A), sends two pulses of light directed toward the two opposite
sides of the wagon. These pulses are emitted exactly at the same time just
when the train is crossing a woman standing on the station (Observer
B). The two observers can synchronize their clocks right at
this time. The question is, how do they observer the events corresponding to
the pulses arriving to the photocells located at the end of the train car?
Notice that the length of the car is known to
be L*.

Newtonian Mechanics
Analysis:

Point of view of Observer
A:

This observer does not even
know, necessarily, that the train is moving implying that he will assign to
both rays the same velocity c. Thus, the distance to be traveled by
the rays is the same and equal to
.
Therefore, the time taking by each ray is

and implying
.

Notice that both rays reach their targets at the same
time or simultaneously.

Point of view of Observer
B:

Classically, Ray 1
travels the distance
plus
the distance that the front of the car moves ahead during the time taken by
the ray in reaching its target,
.
The total distance is
.
The speed at which this ray moves is
.
Thus, the time taken can be calculated from

Ray 2 travels the
distance
minus
the distance moved by the back of the car during the time taken by the ray
in reaching its target,
.
The total distance is
.
The speed at which this ray moves is
.
Thus, the time taken is

Therefore,
which
agrees with the result obtained by the observer in the train.

Semi-relativistic Mechanics Analysis:

Point of view for Observer
A:

Just as in the classical
case, this observer does not even know, necessarily, that the train is
moving implying that he will assign to both rays the same velocity c.
Thus, the distance to be traveled by the rays is the same and equal to
.
Therefore, the time taking by each ray is

and
implying
.

Notice that both rays reach their targets at the same
time or simultaneously.

Point of view of Observer
B:

In coincidence with the
previous analysis, Ray 1 travels the total distance
.
However, the speed of light is unique for all observers, c. Thus, the
time taken can be calculated from the relation

Ray 2 travels the total distance
again
at the speed of light, c. Therefore, the equation for the time taken
can be written as

Therefore,
which
implies that these events are not simultaneous for the observer standing on
the station. Notice that for the observer in the train the events are
simultaneous.

* Notice that this calculation does not
account for any change in length associated to the motion. The purpose of
the present calculation is to open the discussion for the derivation of such
calculations.

Relativistic
Mechanics Analysis:

The most
significant change in the analysis comes from the fact that the observer on the
station does not measure a length L for the train car. In fact,
Observer B measures an Improper Length.
Observer A measures a Proper length. In contrast to Observer A,
Observer B cannot use a meter stick to determine the length of the car.
Thus,

Observer A
as well as Observer B cannot measure the time for these events with a
single clock. Therefore, neither of the two
observers, A and B, can measure a proper time and both times are
improper. Observers traveling with the individual rays will be able to
measure proper times implying that not a single observer can measure proper time
for both rays.

Point of view for Observer A:

Just as in the
classical and semi-relativistic case, this observer does not even know,
necessarily, that the train is moving implying that he will assign to both rays
the same velocity c. Thus, the distance to be traveled by the rays is the
same and equal to
.
Therefore, the time taking by each ray is

and
implying
.

Notice that both rays reach their
targets at the same time or simultaneously.

Point of view of Observer B:

In coincidence
with the semi-relativistic analysis, Ray 1 travels the total distance
.
However, the speed of light is unique for all observers, c. Thus, the
time taken can be calculated from the relation

Ray 2 travels the total
distance
again
at the speed of light, c. Therefore, the equation for the time taken can
be written as

Therefore,
which
implies that these events are not simultaneous for the observer standing on the
station. Notice that for the observer in the train the events are simultaneous.